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Chapter 3: Problem 27

Find the derivative of the following functions by first expanding the expression. Simplify your answers. \(h(x)=\left(x^{2}+1\right)^{2}\)

Short Answer

Expert verified
Answer: The derivative of the function \(h(x) = (x^2 + 1)^2\) is \(h'(x) = 4x^3 + 4x\).

Step by step solution

01

Expanding The Expression

First, we'll expand \((x^2+1)^2\) using the formula \((a+b)^2 = a^2+2ab+b^2\). So for the given expression: \((x^2+1)^2= (x^2)^2 + 2(x^2)(1) + (1)^2 = x^4+2x^2+1\)
02

Finding The Derivative Of The Expanded Expression

Next, we'll find the derivative of \(x^4+2x^2+1\) with respect to \(x\). Use the power rule for each term: \(\frac{d}{dx}(x^4+2x^2+1) = \frac{d}{dx}(x^4) + \frac{d}{dx}(2x^2) + \frac{d}{dx}(1)\) \(\frac{d}{dx}(x^4) = 4x^3\) \(\frac{d}{dx}(2x^2) = 4x\) \(\frac{d}{dx}(1) = 0\)
03

Combine The Derivatives

Now, simply combine the derivatives calculated above to find the derivative of \(h(x)\): \(h'(x) = 4x^3 + 4x\) So, the derivative of the given function \(h(x)=\left(x^{2}+1\right)^{2}\) is \(h'(x) = 4x^3+4x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expansion of Expressions
Expanding expressions is a key algebraic skill. It involves transforming an equation into a more workable form. For the function \(h(x) = (x^2 + 1)^2\), we need to expand it to make differentiation easier.
We use the identity \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = x^2\) and \(b = 1\).
  • Square the first term: \((x^2)^2 = x^4\).
  • Multiply both terms and double it: \(2(x^2)(1) = 2x^2\).
  • Square the last term: \(1^2 = 1\).
Put it all together: \(x^4 + 2x^2 + 1\).
This expanded expression simplifies our next steps by changing a single complex term into a sum of simpler terms.
Power Rule
The power rule is a straightforward method for finding derivatives of terms like \(x^n\). It's expressed as \(\frac{d}{dx}(x^n) = nx^{n-1}\). This means you multiply by the exponent and subtract one from the exponent.

In our function's expanded form, we apply the power rule as follows:
  • For \(x^4\), we get \(4x^3\) because \(n = 4\).
  • For \(2x^2\), we focus on the \(x^2\) part. By multiplying \(2\) with the derivative of \(x^2\), which is \(2x\), we obtain \(4x\).
  • The term \(1\) is a constant, leading to a derivative of \(0\).
The power rule allows us to systematically differentiate each term of a polynomial function.
Function Differentiation
Differentiation lets us find how a function changes. After expanding and using the power rule, we brought together each derivative component.
Here鈥檚 how it applied to our function:

By combining the derivatives of expanded terms \((x^4 + 2x^2 + 1)\), we add them up:
  • \(4x^3\) from \(x^4\)
  • \(4x\) from \(2x^2\)
  • \(0\) from the constant \(1\)
This gives us \(h'(x) = 4x^3 + 4x\).
Differentiation helps us understand the rate of change of \(h(x)\) at any point and is crucial for analyzing the behavior of functions in calculus.

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Most popular questions from this chapter

Special Product Rule In general, the derivative of a product is not the product of the derivatives. Find nonconstant functions \(f\) and \(g\) such that the derivative of \(f g\) equals \(f^{\prime} g^{\prime}\).

Means and tangents Suppose \(f\) is differentiable on an interval containing \(a\) and \(b,\) and let \(P(a, f(a))\) and \(Q(b, f(b))\) be distinct points on the graph of \(f\). Let \(c\) be the \(x\) -coordinate of the point at which the lines tangent to the curve at \(P\) and \(Q\) intersect, assuming that the tangent lines are not parallel (see figure). a. If \(f(x)=x^{2},\) show that \(c=(a+b) / 2,\) the arithmetic mean of \(a\) and \(b\), for real numbers \(a\) and \(b\) b. If \(f(x)=\sqrt{x},\) show that \(c=\sqrt{a b},\) the geometric mean of \(a\) and \(b,\) for \(a>0\) and \(b>0\) c. If \(f(x)=1 / x,\) show that \(c=2 a b /(a+b),\) the harmonic mean of \(a\) and \(b,\) for \(a>0\) and \(b>0\) d. Find an expression for \(c\) in terms of \(a\) and \(b\) for any (differentiable) function \(f\) whenever \(c\) exists.

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals. . A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Use implicit differentiation if needed to find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=c x^{2} ; x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) \(f(x)=\frac{1}{x}\)

Electrostatic force The magnitude of the electrostatic force between two point charges \(Q\) and \(q\) of the same sign is given by \(F(x)=\frac{k Q q}{x^{2}},\) where \(x\) is the distance (measured in meters) between the charges and \(k=9 \times 10^{9} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\) is a physical constant (C stands for coulomb, the unit of charge; N stands for newton, the unit of force). a. Find the instantaneous rate of change of the force with respect to the distance between the charges. b. For two identical charges with \(Q=q=1 \mathrm{C},\) what is the instantaneous rate of change of the force at a separation of \(x=0.001 \mathrm{m} ?\) c. Does the magnitude of the instantaneous rate of change of the force increase or decrease with the separation? Explain.
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